📚 Solving Simultaneous Linear Equations | 解二元一次方程组
Simultaneous linear equations are one of the core algebra topics in IGCSE Mathematics. They require you to find a pair of values that make two linear equations true at the same time. This topic is assessed regularly in both Core and Extended papers, often embedded in word problems and graphical questions.
二元一次方程组是 IGCSE 数学代数部分的核心考点之一。它要求你找到一对数值，使两个一次方程同时成立。该主题在 Core 和 Extended 试卷中经常出现，并且常与文字题和图像题结合考查。
This revision guide explains the graphical meaning, the elimination method, the substitution method, special cases, word problems and common errors. Worked examples are included to help you build confidence for the exam.
本复习指南将讲解方程组在图形上的意义、消元法、代入法、特殊情况、文字题和常见错误。文中配有详细例题，帮助你建立考试信心。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation is a set of two or more equations that share the same variables. In IGCSE mathematics, the linear case usually involves two equations in two unknowns, often written as x and y. You need to find the values of x and y that satisfy both equations at the same time.
联立方程是指两个或两个以上含有相同变量的方程。在 IGCSE 数学中，线性情形通常涉及两个未知数 x 和 y 的两个方程。你需要找出同时满足两个方程的 x 和 y 的值。
For example, the pair below is a system of two simultaneous linear equations:
例如，下面这对方程就是一个由两个一次方程组成的联立方程组：
2x + y = 7
x − y = 2
ve only one equation, because a single linear equation in two unknowns has infinitely many solutions.
答案写成一组有序数对：x = 3，y = 1。只解一个方程是不够的，因为两个未知数的一个一次方程通常有无穷多组解。
2. The Graphical Meaning | 图像上的意义
Every linear equation in x and y can be represented as a straight line on the coordinate plane. When you solve two simultaneous linear equations, you are looking for the point where the two straight lines intersect.
每一个关于 x 和 y 的一次方程都可以在坐标平面上表示为一条直线。当你解两个一次方程组成的方程组时，实际上是在寻找两条直线的交点。
The x-coordinate and y-coordinate of the intersection point are exactly the solution to the system. If the lines intersect at one point, the system has one unique solution. If the lines are parallel, there is no solution. If the lines overlap completely, there are infinitely many solutions.
交点的 x 坐标和 y 坐标正是方程组的解。如果两条直线相交于一点，方程组就有唯一解；如果两条直线平行，则无解；如果两条直线完全重合，则有无穷多解。
In an exam, a graphical question may ask you to plot both lines and read off the intersection point. Algebra methods, however, usually give a more precise answer.
在考试中，图像题可能要求你画出两条直线并读出交点。但代数方法通常能给出更精确的答案。
3. The Elimination Method: Equal Coefficients | 消元法：系数相等
The elimination method works by adding or subtracting the two equations to remove one variable. If the coefficients of one variable are already equal in size, this method is very fast.
消元法的原理是通过将两个方程相加或相减来消去一个未知数。如果其中一个未知数的系数已经相等，这种方法会非常快捷。
Consider the system:
考虑下面这个方程组：
2x + y = 7
x − y = 2
The y terms are +y and −y. They have the same size but opposite signs, so adding the equations will eliminate y.
两个方程中的 y 项分别是 +y 和 −y。它们大小相同但符号相反，因此将两个方程相加即可消去 y。
(2x + y) + (x − y) = 7 + 2
3x = 9
x = 3
riginal equation. Using x − y = 2 gives 3 − y = 2, so y = 1. The solution is x = 3, y = 1.
将 x = 3 代回任意一个原方程。使用 x − y = 2 得到 3 − y = 2，所以 y = 1。答案是 x = 3，y = 1。
Key rule: if the signs are opposite, add the equations. If the signs are the same, subtract the equations.
核心法则：如果符号相反，就相加；如果符号相同，就相减。
4. The Elimination Method: Multiplying First | 消元法：先乘系数
In many systems, the coefficients of x or y are not equal. You must multiply one or both equations by suitable numbers to make a pair of coefficients match.
在许多方程组中，x 或 y 的系数并不相等。你必须给一个或两个方程乘以适当的数，使某一对系数相等。
Solve the system:
解下面的方程组：
3x + 2y = 12
2x + y = 7
Multiply the second equation by 2 so that the y coefficient becomes 2y, matching the first equation.
将第二个方程乘以 2，使 y 项变为 2y，与第一个方程的 y 项系数相等。
4x + 2y = 14
Now subtract the first equation from this new equation:
现在用这个新方程减去第一个方程：
(4x + 2y) − (3x + 2y) = 14 − 12
x = 2
Substitute x = 2 into 2x + y = 7 to get 4 + y = 7, so y = 3. The solution is x = 2, y = 3.
将 x = 2 代入 2x + y = 7，得到 4 + y = 7，所以 y = 3。答案是 x = 2，y = 3。
Always multiply carefully and remember to multiply every term in the equation, including the constant on the right-hand side.
乘法时一定要仔细，并且记住要给方程中的每一项都乘，包括右边的常数项。
5. The Substitution Method | 代入法
s the subject, such as y = 2x − 1. You simply replace that variable in the other equation.
当一个方程已经把 y 或 x 单独放在一边时，例如 y = 2x − 1，代入法尤其方便。你只需将另一个方程中的对应变量替换掉。
Solve the system:
解下面的方程组：
y = 2x − 1
3x + y = 14
Substitute y = 2x − 1 into the second equation:
将 y = 2x − 1 代入第二个方程：
3x + (2x − 1) = 14
5x − 1 = 14
5x = 15
x = 3
stitute x = 3 back into y = 2x − 1 to find y = 2(3) − 1 = 5. The solution is x = 3, y = 5.
再将 x = 3 代回 y = 2x − 1，得到 y = 2(3) − 1 = 5。答案是 x = 3，y = 5。
Substitution is often longer than elimination when both equations are in standard form, but it is very reliable when one coefficient is 1.
当两个方程都是标准形式时，代入法通常比消元法更长，但当某个系数为 1 时，代入法非常可靠。
6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every system has exactly one solution. If the two lines are parallel, the system has no solution. If the two equations describe the same line, there are infinitely many solutions.
并不是每个方程组都恰好有一组解。如果两条直线平行，方程组无解；如果两个方程描述的是同一条直线，则有无穷多组解。
, the system below has no solution because the lines x + y = 2 and x + y = 5 are parallel:
例如，下面的方程组无解，因为直线 x + y = 2 与 x + y = 5 平行：
x + y = 2
x + y = 5
ions gives 0 = −3, which is impossible. This contradiction tells you there is no solution.
两个方程相减得到 0 = −3，这是不可能的。这个矛盾说明方程组无解。
below has infinitely many solutions because 2x + 2y = 4 is just x + y = 2 multiplied by 2:
下面的方程组有无穷多组解，因为 2x + 2y = 4 只是 x + y = 2 乘以 2 的结果：
2x + 2y = 4
x + y = 2
uation, both equations become identical, so any point on the line x + y = 2 is a solution.
当你化简第一个方程后，两个方程完全相同，因此直线 x + y = 2 上的任意一点都是方程组的解。
curs when a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Infinite solutions occur when all three ratios are equal.
利用系数比 a₁/a₂、b₁/b₂ 和 c₁/c₂，当 a₁/a₂ ≠ b₁/b₂ 时有唯一解；当 a₁/a₂ = b₁/b₂ ≠ c₁/c₂ 时无解；当三个比都相等时有无穷多解。
7. Word Problems Leading to Simultaneous Equations | 实际应用题
Many IGCSE exam questions hide simultaneous equations inside real-life situations. Your first job is to define variables clearly, then translate the words into two equations.
许多 IGCSE 考试题把联立方程隐藏在真实情境中。你的首要任务是清楚地设出变量，然后把文字转化为两个方程。
For example, one pen and one pencil cost $0.90, while 3 pens and 2 pencils cost $2.30. Let x be the cost of one pen and y be the cost of one pencil.
例如，一支钢笔和一支铅笔共 0.90 美元，3 支钢笔和 2 支铅笔共 2.30 美元。设 x 为一支钢笔的价格，y 为一支铅笔的价格。
x + y = 0.90
3x + 2y = 2.30
Multiply the first equation by 2:
将第一个方程乘以 2：
2x + 2y = 1.80
Subtract this from the second equation:
用第二个方程减去这个方程：
(3x + 2y) − (2x + 2y) = 2.30 − 1.80
x = 0.50
Then y = 0.90 − 0.50 = 0.40. So one pen costs $0.50 and one pencil costs $0.40.
然后 y = 0.90 − 0.50 = 0.40。所以一支钢笔 0.50 美元，一支铅笔 0.40 美元。
Always write the final answer in the context of the question, with units if necessary.
最后一定要根据题意写出答案，必要时带上单位。
8. Checking Your Solution | 检验答案
After finding x and y, you should substitute both values into the original equations. If the left-hand side equals the right-hand side in both equations, your solution is correct.
求出 x 和 y 后，应把两个值代回原方程。如果两个方程的左边都等于右边，说明你的答案正确。
For the solution x = 4, y = −1 in the system 4x + y = 15 and 2x − y = 9, check as follows:
对于方程组 4x + y = 15 和 2x − y = 9 的解 x = 4，y = −1，可以这样检验：
4(4) + (−1) = 16 − 1 = 15
2(4) − (−1) = 8 + 1 = 9
Both equations are satisfied, so the solution is correct. This habit can prevent careless mistakes, especially when negative numbers are involved.
两个方程都成立，所以答案正确。这个习惯可以避免粗心错误，尤其是在涉及负数时。
9. Common Mistakes to Avoid | 常见错误
Students often lose marks on simultaneous equations because of sign errors. For example, subtracting (2x − y) means subtracting 2x and subtracting −y, which becomes adding y. Misplacing a minus sign can change the whole solution.
学生在解联立方程时常常因为符号错误而丢分。例如，减去 (2x − y) 意味着减去 2x，再减去 −y，也就是加上 y。漏掉负号会改变整个答案。
Another common mistake is multiplying one side of the equation only. If you multiply an equation by 3, you must multiply every term on both sides by 3. Forgetting the constant term is a frequent error.
另一个常见错误是只乘方程的一边。如果将方程乘以 3，必须把两边每一项都乘以 3。漏乘常数项是非常常见的错误。
Some students find x correctly but then substitute it incorrectly into the wrong equation. Always choose the simpler equation for substitution, and write each step clearly.
有些学生算出了 x，却代入了错误的方程。最好选择较简单的方程进行回代，并且每一步都写清楚。
- Check signs when subtracting equations. / 方程相减时要仔细核对符号。
- Multiply every term on both sides. / 两边每一项都要乘。
- Substitute back into an original equation, not a derived one. / 回代到原始方程，而不是变形后的方程。
- Give the solution as x = ..., y = ... / 答案要写成 x = ...，y = ...。
10. Exam-Style Worked Example | 考试型例题解析
Solve the following simultaneous equations:
解下列联立方程组：
2x + 3y = 13
5x − 2y = 4
We can eliminate y by multiplying the first equation by 2 and the second equation by 3. This gives matching y coefficients of 6y and −6y.
我们可以通过将第一个方程乘以 2，第二个方程乘以 3 来消去 y。这样 y 项分别变为 6y 和 −6y，系数大小相等、符号相反。
